## Leibnitz’s theorem

Leibnitz’s theorem, also known as the product rule for differentiation of two functions, is a formula that allows us to find the nth derivative of a product of two functions.

Leibnitz’s theorem is a powerful tool for calculating higher order derivatives. It can be extended to find the nth derivative of a product of more than two functions, by applying the formula recursively.

Leibnitz’s theorem has many applications in mathematics and physics, including the study of partial differential equations, fluid dynamics, and quantum mechanics. It is also used in calculus and analysis to prove many important theorems, such as the mean value theorem and Taylor’s theorem.

#### Note

## Topic

## Problems

- Find the n th derivative of x^2 e^x
- Find the n th derivative of x^2 sin^2 x
- Find the n th derivative of x^2 log 4x
- Prove that (1-x^2) y_2 – xy_1 = 2 if y = (sin inverse (x))^2, apply Leibnitz’s theorem to find n^th derivative
- If tan y=x, then prove that (i) (1+x^2) y_2+2xy_1=0 (ii) (1+x^2) y_ (n+2) +2(n+1) xy_ (n+1) +n (n+1) y_n=0
- If x = tan(log y), find the value of (1+x^2) y_ (n+1) + (2nx-1)y_n + n (n-1) y_ (n-1)
- If y=sin(log(x^2+2x+1)), prove that (x+1)^2 y_(n+2)+(2n+1)(x+1) y_(n+1)+(n^2+4 )y_n=0
- If y=(x+sqrt(x^2-1))^m, prove that (x^2-1) y_(n+2)+(2n+1)xy_(n+1)+(n^2-m^2 ) y_n=0
- If sin^(-1)y=2 log(x+1), prove that (x+1)^2 y_(n+2)+(2n+1)(x+1) y_(n+1)+(n^2+4) y_n=0
- If y=cos(mlog x), prove that x^2 y_(n+2)+(2n+1)xy_(n+1)+(m^2+n^2 ) y_n=0
- If x=sint, y=sin(pt), prove that (1-x^2) y_ (n+2)-(2n+1) xy_ (n+1) + (p^2-n^2) y_n=0
- If y^(1/m)+y^(-1/m)=2x, prove that (x^2-1 )y_(n+2)+(2n+1)x y_(n+1)+(n^2-m^2 ) y_n=0